JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A10305, doi:10.1029/2005JA011042, 2005
Application of linear regularization methods to Arecibo vector velocities Michael P. Sulzer, Ne´stor Aponte, and Sixto A. Gonza´lez Arecibo Observatory, Arecibo, Puerto Rico Received 2 February 2005; revised 2 June 2005; accepted 14 July 2005; published 19 October 2005.
[1] Estimates of the three-dimensional ion velocity field can be difficult to make with
monostatic radars because there are three unknown components for each independent line of sight velocity measurement. To cope with this problem, one or more assumptions about the vector field must be made to arrive at a solution. At Arecibo, one can measure the ion vector velocities by continuously rotating the antenna beam back and forth 360 degrees in azimuth at 15 degrees off zenith to sample the horizontal components. Until recently, the line of sight velocities obtained from this experiment were typically converted into vector velocities by assuming that the vector field remains constant during one rotation and that horizontal gradients are negligible. In this paper we show how to apply the linear regularization inversion method to the problem of computing ion vector velocities. This technique improves the accuracy of the vector velocities obtained from the measured F region line of sight velocities and also provides a convenient way to do the computations for dual-beam experiments. The technique could improve the vector velocities at other monostatic incoherent scatter radar facilities. Citation: Sulzer, M. P., N. Aponte, and S. A. Gonza´lez (2005), Application of linear regularization methods to Arecibo vector velocities, J. Geophys. Res., 110, A10305, doi:10.1029/2005JA011042.
1. Introduction [2] Incoherent scatter radars (ISR) provide a powerful tool to measure the ion motion along the radar line of sight. In most cases one is more interested in knowing the full velocity field, particularly the motion along the magnetic field lines, which provides information about neutral winds and diffusion, while the ion motion across field lines is a measurement of electric fields. In order to measure the three orthogonal components of the vector velocity, one would need either three beams pointing simultaneously at the same volume or a monostatic instrument with steering capabilities. In the case of a monostatic radar, a vector solution is constructed from measurements taken at different times and locations, so one needs to make assumptions about the temporal and spatial variation of the vector field as the instrument completes a cycle. [3] For more than 3 decades, the Arecibo ISR has provided measurements of ion vector velocities by using the technique developed by Hagfors and Behnke [1974]. This method takes advantage of the azimuth steering capabilities of Arecibo so that incoherent scatter (IS) autocorrelation functions are measured while rotating the line feed antenna continuously in azimuth at 15 degrees off zenith. The line of sight velocities obtained from the IS autocorrelation functions then get converted into vector velocities by using a least squares fitting (LSF) approach with the assumptions of a constant vector field in one Copyright 2005 by the American Geophysical Union. 0148-0227/05/2005JA011042
rotation and no horizontal gradients. This technique yields independent vector velocities every 15 min (the time of one azimuth rotation) for each altitude sampled in the F region. The 15 min resolution turns out to be adequate for quiet time conditions, but at times when there are rapid changes in the ionosphere, it may not be sufficient to represent the variations in the vector velocities. [4] The measurement of ion velocities at Arecibo improved significantly in the mid-1980s with the introduction of the ‘‘multiradar’’ experiment developed by Sulzer [1986]. With this new mode, the statistics of the p incoherent scatter ffiffiffi spectra were improved by a factor of 7 (2.65), thus reducing random errors in the line of sight velocities by the same factor. Of course, more accurate line of sight velocities directly translate into more accurate estimates of the vector velocities. Moreover, there is a subtle effect (see section 6) in which random errors introduce a nonphysical anticorrelation between different components of the motion, so reducing random errors also mitigates this effect. [5] Following the upgrade of the IS spectral measurements, Burnside et al. [1987] modified the method of Hagfors and Behnke [1974] to allow for the possibility of constant horizontal gradients. Their study found mostly negligible gradients in the horizontal plane dvy dvx (h i 0, h i 0) but a small horizontal gradient dx dy dvz , which can reach 0.1 m/s/km in the vertical velocity dx and lead to a significant error of about 30 m/s in vx if not taken into account. The largest gradient is dvz the vertical gradient in the upward component ( ) dz
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attributed to variations in diffusion along B. The vertical variations could also be measured with the LSF method since vector velocities are computed as a function of altitude. The technique of Burnside et al. [1987], known as the ‘‘constant gradient’’ method, still assumes no time variations in the vector field during one rotation. [6] The assumption of a vector field that remains constant for 15 min as the Arecibo antenna rotates in azimuth can be somewhat restrictive at times when the ionosphere over Arecibo experiences disturbances. In fact, we will see in section 4 how some rapid variations in the vector components cannot be resolved accurately with the LSF method. The limitations of the LSF technique along with the desire of a convenient method for dual-beam measurements prompted us to explore new alternatives to get the vector velocities at Arecibo. A very convenient and powerful approach consists of using linear regularization inversion methods [Press et al., 1992]. The linear regularization machinery provides a general algorithm to add information to the problem so that it becomes well-posed. The main assumption that goes into the method is that vector velocities vary linearly between time steps. This information is added to the problem in the form of a matrix H and its strength is controlled by a single parameter l, as described in section 3. [7] Following the description of the technique in section 3, we show in section 4.1 that the regularization process is capable of performing the inversion with sufficiently small errors. We must show this, although we expect it to work for these measurements since we already know that the inversion works using least squares fitting. We thus find a reasonable value for the so-called regularization parameter and proceed in section 4.2 to study the errors in the inversion process using this parameter value. There are statistical errors, of course, but also there is some bias, which shows up as a delay in responding to rapid temporal changes in the velocities. Such bias is inevitable when one does not have an accurate model of the underlying process. Section 5 presents a comparison between simulated results using regularization and LSF. The advantage of the regularization is apparent for reducing the bias, that is, making more accurate measurements with a rapidly changing underlying process. Section 6 analyzes the correlation between the errors in two of the components and shows that the regularization gives superior behavior in reducing the effects of this correlation. Finally, section 7 presents the first dual-beam measurements. The dual-beam configuration increases greatly the sensitivity in the measurement of horizontal velocities and also adds the possibility to measure directly any possible horizontal gradients in scalar parameters like electron density and temperatures. In this study we also show that the new regularization technique can be easily extended for dual-beam measurements.
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F region experiment for vector velocities samples about 15 heights (37 km resolution) while rotating back and forth in azimuth (f). Each line of sight measurement is a superposition of three orthogonal components that can be written in matrix form as 2 4
1 VLOS
3
2
5¼4
2 VLOS
cos f sin q1
sin f sin q1
cos f sin q2
sin f sin q2
2 3 vx 3 7 cos q1 6 6 7 56 vy 7 6 7 cos q2 4 5 vz
where each line of sight (VLOS(h, t)) is a function of height and time, while the angle q is the zenith angle. The coefficients [vx vy vz]T are the unknown vector components in the south, east, and upward direction, respectively, and superscripts 1 and 2 stand for first and second beam, respectively. With one beam, there are three unknowns for each measurement, so obviously the problem is underdetermined. With two beams things improve, but the problem remains underdetermined, and some assumptions must be made in order to get a unique solution. [9] The above coordinate system is useful to visualize the actual motion in a coordinate system that we naturally relate to. However, to get a handle on the electric fields and motion parallel to the Earth’s magnetic, we must transform the geographical velocities to a coordinate system aligned with the Earth’s magnetic field. At Arecibo where the dip angle is about 45 degrees, a useful coordinate system is the one shown in Figure 2 where ^x is up along B and ^y is perpendicular to B and east and ^z is upward and northward also perpendicular to B. [10] With the geomagnetic coordinate system of Figure 2, the perpendicular to B ion drifts and the ion velocity along the magnetic field can be written in terms of the geographical velocities, the dip angle I, and the declination d as follows: 2
Vpn
2
3
cos d sin I
6 7 6 6 7 6 6 Vpe 7 ¼ 6 sin d 6 7 6 4 5 4 Vpar cos d cos I
sin d sin I cos d sin d cos I
cos I
32
vx
3
76 7 76 7 6 7 0 7 76 vy 7: 54 5 sin I vz
3. Linear Regularization [11] The technique developed by Hagfors and Behnke [1974] overcomes the lack of information on each measurement by using many samples, while assuming that the unknowns do not change in the time of one rotation. Such scheme leads to an equation system (one beam)
2. Velocity Field From Ion Line of Sight Velocity [8] In Figure 1 we show the geographical coordinate system adopted in this study to represent the Arecibo velocity measurements. A right-hand coordinate system can be obtained by aligning the x-axis with the south direction and the y-axis with the east. A typical dual-beam 2 of 11
2
VLOS ð1Þ
6 6 6 6 4
.. . VLOS ðnÞ
3
2
cos f1 sin q
7 6 7 6 7¼6 7 6 5 4
sin f1 sin q .. .
cos fn sin q
sin fn sin q
cos q
32
vx
3
76 7 76 7 76 vy 7 76 7 54 5 vz cos q
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and a quadratic. The extra information from any of these function enters into the problem as an additional matrix so that the new system of equations can be written as
AT A þ lH v ¼ AT b;
ð3Þ
which in appearance differs from the least squares fitting method (equation (1)) by the term lH. In practice, when we implement equation (3) for the Arecibo vector velocities, the dimensions of each matrix are different from the LSF method. For each height we now solve equation (3) by using measurements from a whole experiment (from a few hours to a few days) instead of an azimuth rotation. The new A matrix has the form 2
6 6 6 0 6 A¼6 6 6 6 4
Figure 1. Geographic coordinate system including dualbeam. which is of the form b = A v. This system has n measurements and only three unknowns (n > 3), so it can be solved for v using least squares fitting AT Av ¼ AT b:
ð1Þ
In a similar manner, the general inversion problem considers the estimation of an unknown quantity v from a set of measurements b by some minimization principle, like [Press et al., 1992] c2 ¼ j A v bj2 ;
ð2Þ
the matrix A may have fewer rows than columns (i.e., more unknowns than measurements), in which case the solution is not unique. The solution can be made unique by adding extra information or some constraint, in the manner described by Press et al. [1992] under linear regularization. The technique is quite general, since it allows for arbitrary functions to be added as the regularization or smoothing part. Examples of functions are a constant (i.e., the variation of the unknowns between time steps is quite smooth), a line,
c11
0
c12
c13
0
0
0
0
0
c24
c25
c26
0
0
cl3l2
cl3l1
.. 0
0
0
7 7 7 7 7 7 7 7 5
.
3
cl3l
where each row has three nonzero terms, cij = cos fi sin q, cij+1 = sin fi sin q, and cij+2 = cos q, and dimensions N 3N, where N is the number of line of sight measurements used in the solution (very flexible since we could use any data segment, but usually the whole experiment). The vector b has the line of sight measurements b ¼ ½VLOS ð1Þ VLOS ð2Þ . . . VLOS ð N ÞT :
The vector of unknowns is of length 3N
T v ¼ vx ð1Þ vy ð1Þ vz ð1Þ vx ð N Þ vy ð N Þ vz ð N Þ :
Each row of the matrix A and measurements vector b must be scaled by the corresponding measurement error (standard deviation of VLOS). [12] Finally, we should comment on the actual form of the matrix H. Through this matrix, we add a priori information that allows us to get a unique set of vector velocities at once. The main assumption to solve the problem is that the vector
Figure 2. Geomagnetic coordinate system. 3 of 11
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Figure 3. Vector velocity input-output comparison using a simulation assuming a single beam and applying the method of regularization. The panels on the left show the results for a small value of l. The panels on the center were obtained from a simulation with an optimal value of l, while the panels on the right correspond to a case with too much l. velocities vary linearly between time steps. These time steps can be as short as 10 s for F region IS spectral measurements and longer (about a minute) when running multiple experiments. For a 1 min time step we can safely assume a linear variation between adjacent points. The general matrix H for a linear variation between time steps is given by Press et al. [1992, equation [18.5.12]]. Here we have modified the H matrix slightly, since we must apply the same constraint to three different components. The reader interested in the details about this matrix is referred to Appendix A. [13] By taking advantage of the sparse nature of the matrices, a code has been implemented at Arecibo which solves equation (3) very efficiently through the banded matrix formulation of Press et al. [1992]. Using this code, it is possible to compute the ion vector velocities for an entire World Day (a few days worth of data) in a matter of seconds (assuming a constant sLOS). Moreover, comparable speeds can be achieved in a high-level interpreted language like MATLAB, by using its sparse matrix definitions.
4. Performance [14] In this section we must deal with two important issues related to the implementation of linear regularization to compute Arecibo vector velocities. First of all, we must determine a reasonable value of the parameter l for the problem at hand. After finding this value, we then need to determine what are the expected errors in each of the velocity components. 4.1. Optimal Value of L [15] In order to find a reasonable value for the parameter l, we have used a simple simulation scheme. We begin by generating a time series of vector velocities. This known vector field (v) becomes the input to the simulation in which we convert these values into line of sight velocities (VLOS = Av) by assuming that the Arecibo beam points 15 off zenith while the antenna rotates continuously in azimuth. The line of sight velocities then become our data after adding random noise. Once we have a synthetic data set, we can apply the regularization inversion technique to estimate the ion vector velocities over the field of view of the radar. Now from equation (3), the regularization method requires both the matrix H and the parameter l. The matrix H is determined by our assumptions (linear variation between
time steps), but the parameter l has to be estimated somehow, in our case from the simulations. The idea is that for each l, the simulation computes a new set of vector velocities. This operation is then repeated for many values of l to get statistics; in particular we can form estimates of the errors in each component by comparing the results from each run vi(l) to the original vector components vi s2v ðlÞ ¼
n 1X ðvi vi ðlÞÞ2 : n i¼1
[16] A great advantage of using the simulation is that we can easily see how the parameter l affects the solution when using the synthetic data set as input. Examples of the quantities involved in the simulation are shown in Figure 3. The figure has results from the simulations with three different values of l. On each set, the top panel shows the typical variation of the azimuth angle for a beam swinging experiment in which the antenna continuously rotates back and forth by 360. The second panel shows the line of sight velocity (actually two curves). One of the LOS time series corresponds to our synthetic data obtained from the three components of the ion vector velocity (v) represented by the dark smooth lines on the next three panels. These smooth vector components are our input vector velocities obtained by assuming that the velocities are primarily constant except for 1/2 hour to 1 hour gaussian pulses. The gaussian pulses of 1/2 to 1 hour sort of resemble the signature of traveling ionospheric disturbances (TIDs) going over Arecibo. [17] A good test for the method is that when using the line of sight curve on each second panel, the output should look like the thick smooth curves of the last three panels. Since the solution depends on the value of l, it is necessary to find the range of values of that parameter where the solutions are best. For very small values of l (panels on the left of Figure 3), the effect of regularization is limited. Because the problem is quite underdetermined, it is very easy to find a set of vector velocities that reproduce the ‘‘measured’’ line of sight velocities almost exactly (no difference is seen between the two line of sight curves of the second panel). However, the solution is not in agreement with the original values, and we see that the thin lines of the last three panels are quite unstable and extremely noisy.
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Figure 4. Standard deviation as a function of l for the three velocity components. The curves on the left panel were obtained from the simulations with constant input vector velocities plus gaussian pulses, assuming a single beam. The right panel is like the left panel but for a dual-beam configuration. [18] Next we consider the solution obtained with larger values of l in Figure 3. The center panels show an example where the solution was computed with a value of l that minimizes the errors in each component. In this case, not surprisingly, the agreement between the input vector (thick lines of last three panels) and the output solution (thin light lines) is very good. Although hard to see, the output line of sight velocity (thin line of second panel) turns out to be a smooth version of the input line of sight. The value of l for this particular case was obtained from the error curves shown in Figure 4 (left panel). The scale used for that figure is logarithmic, so actually the curves cover a large range of values of l. In particular, they show how for small l the errors are very large since the solutions are quite unstable as shown in Figure 3. As l increases, the errors come down until they reach a minimum. It is around this minimum where we read the l value used to compute our optimal solution. Note, however, that the three minima do not lie exactly on top of each other, but they are close enough so it is possible to pick up one value that works well for all three components. Since the effect of the parameter l is to smooth out or stabilize the solution, adding more regularization (i.e., increasing l) beyond the point where the minimum is reached leads to solutions that are even smoother but not more accurate. An extreme case is shown in the panels on the right-hand side of Figure 3, where l is so high that the solution cannot respond even to changes of the order of 1 hour. The fact that this solution is extremely smooth makes a noticeable effect in the line of sight velocities corresponding to this set of vectors. A comparison of the two LOS time series in the second panel shows that the new time series (thin lines) differs considerably from the original curve at times of rapid changes. [19] Also shown in Figure 4 (right panel) is a plot of the error in all three components as a function of l computed for the case of having two simultaneous radars (dual-beam) pointing 180 away from each other in azimuth. As expected, the errors obtained with two simultaneous measurements are smaller than the errors obtained with a single beam. For example, at l = 100, the errors for a single beam are 4.9, 7.3, and 4.8 m/s for the perpendicular north, perpendicular east, and parallel components, respectively, while the corresponding values for two beams are
3.6, 4.0, and 3.1 m/s. These numbers are not the actual standard deviations, since they are just based on differences between a particular data set and the true velocities, but from those values we can see that there is a significant improvement in the measurements along the perpendicular east direction. The curves on the right-hand side panel also have a wider minimum than the curves on the left, so there is a larger range of values of l for which the solutions with dual beam have higher accuracy. 4.2. Statistical and Systematic Errors [20] The analysis performed to obtain a reasonable value of l was based on a simulation with a noisy data set and many values of l. With that scheme we got curves with some measure of errors as function of l. However, what we are after are the statistical uncertainties and systematic errors that might be present in the solutions obtained with the good value of l, that is, the behavior of the total error as a function of time (an error bar for each data point). [21] It is standard practice with least squares fitting problems to estimate the statistical uncertainties in the parameters from the covariance matrix. As pointed out by Aster et al. [2005], it is also possible to obtain a covariance matrix for linear regularization problems, but unlike least squares fitting solutions, the linear regularization solutions are biased. The bias can even be larger than the statistical uncertainties obtained from the covariance matrix, so the errors obtained from the covariance matrix have little meaning if the possible biases are unknown. Furthermore, the Arecibo vector velocities computed with either LSF and regularization will have biases, since both techniques introduce smoothing to quantities which vary as a function of time. The smoothing in the LSF comes from assuming constant vector velocities during the 15 min of one rotation, while in regularization the smoothing comes from the stabilizing lH term. The fact that the solutions obtained with either technique are smoothed versions of the true velocities means that there would be a systematic error in the estimates. Thus the total error is a combination of the statistical uncertainties and the systematic errors. [22] To estimate the variance of the vector velocities computed with linear regularization, one needs to compute the covariance matrix, that is, the expected value
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Figure 5. (left) Standard deviation of geomagnetic velocity components as a function of time, obtained from the covariance matrix. (right) Same as left panel but from Monte Carlo simulation. E[(v hvi)(v hvi)T] Cov(v), where v is the velocity vector in equation (3) and the quantity hvi is the mean value. Using the notation of equation (3), it can be shown that the covariance matrix is CovðvÞ ¼
h
AT A þ lH
1
AT
ih
AT A þ lH
1
AT
iT
;
ð4Þ
and from this matrix, the standard deviation is obtained from the square root of the diagonal terms. In the left panel of Figure 5, we show an example of using equation (4) to get the standard deviation of the vector velocities obtained from the LOS velocities in the simulation of Figure 3. It was assumed for this calculation that the LOS velocities had a constant standard deviation of 8.5 m/s and the regularization parameter l was set to 100. The first thing to notice in this plot is that errors are pretty high at the edges for all three components. This behavior is consistent with the fact that the regularization inversion requires information from both the past and the future, and of course, at the edges either the past or the future is missing. As the data collection goes on, we then see that the errors come down and remain oscillating periodically around a certain level. These oscillations in the errors are also not surprising, as they reflect the rotation of the antenna in the beam-swinging mode assumed in the simulation. As expected, the largest errors occur in the perpendicular east component, which is a purely horizontal component that is far less sampled with the beam oriented at 15 degrees off zenith. The mean standard deviations (including the effect at the edges) for this single beam (l = 100) simulation are 6.71, 3.76, and 3.71 m/s for the perpendicular east, perpendicular north, and antiparallel components, respectively. The results obtained from the covariance matrix have been verified with a Monte Carlo simulation. The panel in the right of Figure 5 shows that the standard deviation for each component obtained from the Monte Carlo simulation has basically the same temporal variation and amplitude as the results found with the covariance matrix calculation. [23] The numbers that we have just quoted for the uncertainties in the geomagnetic components are less than 7 m/s for the perpendicular east and less than 4 m/s for the other two components. Yet we see in the center plot of Figure 3 that there are differences between the input and output components that significantly exceed these values.
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Clearly then, the statistical uncertainties are not the whole story, as mentioned earlier. Now, in order to properly determine the biases in the solutions, it is necessary to know the true velocities, which we know for the simulations, as well as the mean value for each component. The mean value is estimated from a Monte Carlo simulation, but for this well-behaved regularization process, it turns out that the mean value is equal to the solution obtained in the absence of random noise in the LOS velocities. As we can see in the first three panels of Figure 6, the noise free vector solutions (labeled Mean) are similar but not equal to the true values of each component. The reason for this is that in going from the input vector velocities to the line of sight velocities (assuming a beam-swinging experiment), 2/3 of the information is lost in the transformation (at each time step there is only one LOS velocity measurement and three unknown vector components). Considering this fact, the inversion process does a very good job, but it is important to keep in mind the great limitation of this problem. In the last three panels of Figure 6 we show the difference between the mean value of each component and the true value. This difference quantifies the bias in the solution. As we can see from this example, the larger biases occur at times of rapid changes in the individual velocity components. In some instances, the bias significantly exceeds the mean statistical uncertainty (plotted as dashed lines in the last three panels). However, for the most part, the bias is of the order of, or less than, the statistical uncertainties for this example with slowly (1/2 hour to 1 hour) varying gaussian pulses.
5. Comparison Between Regularization and LSF [24] The simulations described in the previous section also provide a great tool for comparing the reconstructions obtained from different methods. For this particular test, we use slightly different inputs, so instead of vector velocities that look like constant with gaussian pulses, we now have vector velocities that vary in arbitrary fashion (obtained
Figure 6. Comparison between known input velocities and mean (same as noise free LOS) solution, in the first three panels. The last three panels show the bias for each component.
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Figure 7. Comparison between least-squares fitting method and regularization with sLOS = 8.5 m/s. First column least-squares fitting with 1 beam, second column regularization with 1 beam (l = 100) and third column regularization with 2 beams (l = 100). Perpendicular east first row, perpendicular north second row and parallel third row.
from data taken in the night of 17– 18 February 1999). The idea, however, remains the same; that is, we start with clean vector components and then transform these into line of sight velocities and add some noise to the LOS velocities. Finally, we turn the LOS velocities back into vector velocities by using either LSF or regularization. The results of this comparison are summarized in Figure 7, where we show a plot matrix containing input and output velocities from different methods (columns) for each of the three orthogonal components (rows). Again we are considering the three-dimensional velocities in geomagnetic coordinates; thus the first row corresponds to the perpendicular to B and east component, the second row has the perpendicular to B and north component, and the third row are velocities parallel to B but positive upward (antiparallel). Basically, there are two important things to notice from this figure. First of all, one can see that the perpendicular east (Vpe, first row) is the most difficult component to get right, regardless of the method. As we pointed out earlier, the errors on this component are higher because the radar looks almost vertical (15 off zenith), while this component is completely horizontal. The second important thing to notice is that the results obtained with regularization (second column) are better than LSF (some points of the perpendicular north regularization 1 beam solution have higher bias than LSF, though), since the output curves (thin lines) are closer to the input (thick lines) than with LSF. With the second beam, the results become quite accurate, as shown in the third column, which was computed also with the regularization method. [25] The statistical and systematic errors corresponding to the velocities in Figure 7 are shown in Figure 8. The first row of the figure shows the standard deviation (statistical uncertainties) for LSF and regularization with one and two beams. These values were obtained from the covariance matrix, assuming a LOS velocity standard deviation of
8.5 m/s and using l = 100 in the regularization method. From the first two columns of that first row we see that the standard deviation of the perpendicular east component is about 0.6 m/s smaller with LSF than with one beam regularization, but the opposite is true for the perpendicular north and parallel components where regularization does better than LSF, by about the same amount. Thus we see that as far as the statistical uncertainties is concerned, there is not much gain in going from LSF to regularization with just one beam. The third box of the first row shows that the statistical errors are reduced when using a second beam, especially the perpendicular east component. [26] The great advantage of the linear regularization method over LSF becomes evident when one considers the biases introduced by each method. In the second row of Figure 8 we compare the biases in the perpendicular east component. Neglecting effects at the edges, we can see that basically the biases from linear regularization can be a factor of 2 smaller than LSF in single beam measurements. This improvement is very significant when considering that the bias in this component of the LSF solution can reach a 40 m/s error, which is more than 5 times the statistical uncertainty. The maximum bias of this component with two beams is less than 10 m/s, which is 4 times smaller than one beam with LSF. For the other two components, shown in the third row, we see that the biases are somewhat smaller for the parallel component when using regularization and basically just a little bit higher in the perpendicular north with regularization. The dual-beam case shows that biases in these other two components remain mostly below 5 m/s.
6. Correlation of Errors and Horizontal Motion [27] The presence of random fluctuations in the LOS velocity introduces uncertainty in the determination of the vector velocity. Relative uncertainties in the geomagnetic
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Figure 8. Standard deviation (first row) and bias (second and third rows) comparison between regularization and LSF. See text for description. components were originally computed by Hagfors and Behnke [1974] for different azimuth rotation schemes (e.g., 0 – 360, 0– 180) using the LSF method. In addition, these authors looked at the correlation between random fluctuations of the different components and found that for a rotation scheme 0 – 360, the perpendicular-north and parallel components are almost anticorrelated (hVpar Vpni = 0.9). Graphically, this anticorrelation would be equivalent to ions moving horizontally. [28] Concerning the correlation of random errors in the beam-swinging experiment, Hagfors and Behnke [1974] pointed out that one must be careful in not assigning any geophysical interpretation to this effect. Thus it is important to be able to distinguish between the signature of nonphysical correlations introduced by random noise and the actual motion of the ionosphere. To show the reader how random errors alone can introduce anticorrelation, we have performed a simple simulation (see Figure 9) in which the input LOS velocity is just zero mean gaussian random noise (s = 8.5 m/s). We then solve for the vector velocities assuming a zenith angle of 15 and a 0 –360 back and forth continuous azimuth rotation. The outputs of the simulation shown in Figure 9 are the perpendicular-north and parallel components obtained with LSF (second panel) and regularization (third panel). From the second panel (LSF method), it is quite evident that both components of the motion are nearly equal but opposite in sign. The correlation coefficient for this case is about 0.927, which is close to the result of 0.9 found by Hagfors and Behnke [1974] assuming a dip angle of 50. The third panel (regularization) also shows a strong sign of anticorrelation but somewhat less than LSF, since the correlation coeffi-
cient is about 0.73. Two key characteristics of the anticorrelation introduced by the noise are the correlation time and the amplitude of the fluctuations. In the second panel we see that some of the correlated features can last from 30 min to 1 hour, and the amplitudes can be of the order of 10 m/s. The correlated features observed in the third panel (regularization) are better resolved than with the LSF method, and that leads to somewhat narrower correlation length but similar amplitude. From the results of this simulation, we can conclude that correlations in the Vpn and Vpar components which are shorter than 1 hour and with amplitudes smaller than 10 m/s (near 360 km of altitude) can be purely due to noise. On the other hand, correlated features in those two components that last for several hours and have amplitudes, say, larger than 15– 20 m/s near 360 km can be resolved more accurately by this technique and therefore regarded as having geophysical origin.
7. First Arecibo Dual-Beam F Region Velocities [29] The Arecibo dual-beam radar became operational in the summer of 2001, and the first measurements of F region velocities using this new system took place during the 11 – 15 July 2001 world day. For this particular experiment, the gregorian and line feed antennas were pointing in opposite directions, that is, 180 apart in azimuth, while rotating continuously back and forth (a beam-swinging experiment) at 15 off zenith. In this configuration, when one antenna (i.e., one radar) is probing the ionosphere, say, north of Arecibo, the other is sampling an ionosphere that is south of Arecibo, and the same for east-west and all other directions
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Figure 9. Correlation of random errors for the perpendicular north and parallel ion velocities. First panel shows gaussian random noise as line of sight velocity. Second panel shows perpendicular-north and parallel velocities obtained by processing the LOS velocities with the LSF method, assuming the azimuth rotation in the fourth panel and a 15 zenith angle. Third panel is the same as second but computed with the regularization method.
as the antennas rotate. This configuration can tell us if there are horizontal gradients in electron density, temperature, and other scalar parameters of the medium, but still there is not enough information to determine if the three-dimensional velocity varies within the field of view. However, since one antenna leads the other by about 7.5 min and they look at opposite directions, if the solutions happen to look similar at least we can confirm that the variations during one rotation cannot be large and if there are horizontal gradients these would have to be constant. [30] In Figure 10 we show the line of sight velocity measurements taken in 14 –16 July 2001 in the first two panels and the corresponding vector velocities in the last three panels. The top curve on each of the last three panels comes from measurements obtained with the line feed antenna, the middle curve from the gregorian, and the lower curve corresponds to the solution obtained by combining the measurements from both antennas. The velocities that we see on each panel were computed with the regularization method from the LOS velocities using l = 100. Certainly, it is quite encouraging to see that the velocities obtained with the gregorian antenna follow the same trend as the velocities from the line feed antenna. Furthermore, the agreement extends to some of the shorter timescale, large-amplitude perturbations, as for example, near 0100 on 15 July. Hence
the agreement in the velocities from the two individual beams basically checks that (1) the gregorian systems works as well as the line feed system and that (2) in the time scale of half a rotation (about 7.5 min) the velocities do not change much. [31] The solution obtained by combining the simultaneous LOS measurements from the two beams (bottom curve on each panel) also presents very similar features as the ones in the curves obtained from the individual beams. Thus the regularization technique provides a convenient way to process the dual-beam data. However, without knowing the true velocities it is not possible to say that this dual-beam solution is better than the other two, but in some instances we see more fine structure than with the single beam solutions, which is an indication of the better resolution possible with the new system. [32] Finally, we consider the statistical uncertainties of the geomagnetic velocities from the first Arecibo dual-beam experiment. In Figure 11 we show the time variation of the standard deviation for a subset of the data in Figure 10. In the first box we consider the errors for the velocities obtained from linefeed data only, the second box is from the gregorian, and the third box corresponds to the results obtained by using data from both beams. As we can see, the temporal behavior of the standard deviation is similar to
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Figure 10. Comparison between Arecibo single and dual beam vector velocities. The first two panels show the actual line of sight measurements for both the line feed and gregorian radars. The third panel shows the perpendicular-east component, the fourth panel shows the perpendicular-north, and the fifth panel shows the parallel component. On each of the last three panels, the top line is the velocity obtained from line feed data and displaced by 100 m/s, the middle line is obtained from the gregorian, and the bottom line is the solution after combining data from both feeds and displaced by 100 m/s. what had been found earlier for the velocities from the simulations, for all three components. Thus one can see higher errors at the edges and smaller errors in the middle with oscillations related to the beam-swinging motion. For these calculations we have used the actual LOS standard deviation for each measured point. The mean values of those errors in the LOS were somewhat smaller than what it was used in the previous simulations, that is, 8.13 and 7.65 m/s sLOS for the linefeed and gregorian, respectively, while the earlier simulations were done with a constant sLOS = 8.5 m/s. The lower sLOS leads to smaller errors in all components, with the gregorian values slightly smaller than
the linefeed for this reason. In terms of the numbers, one can see that the statistical uncertainties for single beam measurements once things quiet down are of the order of 6 m/s for the perpendicular east and less than 4 m/s for the other two components. The dual-beam brings those numbers down to 4 m/s for the perpendicular east and less than 3 m/s for the other components. Unfortunately, for actual measurements we cannot estimate the bias in the solutions, since, of course, we do not know the true velocities. However, as we have seen earlier with the simulations, biases in the velocities can be significantly larger than the statistical uncertainties, so the reader is warned that error bars based
Figure 11. Standard deviation of geomagnetic velocity components for a subset of the data taken in 14– 16 July 2001. From left to right, linefeed, gregorian, and dual-beam, respectively. 10 of 11
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on the statistical uncertainties most likely underestimate the total error.
8. Summary [33] Two very significant improvements to the Arecibo vector velocity measurements have taken place with the completion of the dual-beam project and the introduction of a new data processing technique based on linear regularization methods. In this study we have done an extensive evaluation of the performance of linear regularization to compute Arecibo vector velocities. More specifically, in section 4.1 we showed how to find a reasonable value of lambda, the regularization parameter, using simulated data. We described the trade off between smoothness and accuracy, and showed that there are acceptable compromises between stability of the solution and the ability to follow realistic time variations. In section 4.2 we examined statistical and systematic errors, also using simulated data and the reasonable value of lambda. We found that the bias is significant compared to the statistical errors only during times when the velocity changes quickly. The bias shows up as a delay in following the change without overshoot or instability and thus is like smoothing the original velocities. [34] Section 5 makes the comparison between regularization and LSF; the bias of the perpendicular east component is much smaller with regularization, but statistical errors are not very different. Section 6 considered correlation of errors between the perpendicular north and the parallel components. This correlation mimics physically interesting results, and so it is important to show how to distinguish actual correlation in the velocity components from apparent correlation due to the errors. We showed that a large enough magnitude and a long enough persistence demonstrate that the correlation is in the velocities. We presented the first dual-beam measurements. Although the velocities using data from both beams show somewhat more detail than those data using data from either single beam, we can conclude that single beam data is sufficient for these normal conditions. Dual beam measurements provide a margin of safety allowing high accuracy to be maintained during impulsive events. These measurements also show that during conditions known to be normal, one beam could be located at 15 degrees zenith, and the other could be used in a different position, such as vertical, to allow a different kind of measurement (for example, signatures of gravity waves in the vertical electron density) along with the velocity measurements.
Appendix A:
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which gets shifted along the diagonal. In our case the vector v (^ u(x) in the work of Press et al. [1992]) has three components for each time step, so the H matrix in equation (18.5.12) has to be modified to apply the same weights to each component. This task is accomplished by repeating each row of the Press et al. [1992] matrix three times and adding two zeros between each nonzero element, as shown in the matrix that follows. 3 1 0 0 2 0 0 1 0 0 0 0 0 0 0 0 0 6 0 7 1 0 0 2 0 0 1 0 0 0 0 0 0 0 0 6 7 6 0 0 1 0 0 2 0 0 1 0 0 0 0 0 0 0 7 6 7 6 0 5 0 0 4 0 0 0 0 0 0 0 0 0 7 7 6 2 0 6 0 2 0 0 5 0 0 4 0 0 0 0 0 0 0 0 7 6 7 6 0 0 2 0 0 5 0 0 4 0 0 0 0 0 0 0 7 6 7 7 6 1 0 0 4 0 0 6 0 0 4 0 0 1 0 0 0 7 6 7 6 0 1 0 0 4 0 0 6 0 0 4 0 0 1 0 0 6 7 .. 6 7 7 6 . 6 7 6 0 0 0 1 0 0 4 0 0 6 0 0 4 0 0 1 7 7 6 7 6 0 0 0 0 1 0 0 4 0 0 5 0 0 2 0 0 6 7 6 0 0 7 0 0 0 1 0 0 4 0 0 5 0 0 2 0 7 6 6 0 0 7 0 0 0 0 1 0 0 4 0 0 5 0 0 2 6 7 6 0 0 0 0 0 1 0 0 2 0 0 1 0 0 7 7 6 0 0 4 0 0 0 0 0 0 0 0 1 0 0 2 0 0 1 0 5 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 1 2
The matrix above has dimensions 3N 3N, where N is the number of records used to compute a solution (usually the whole experiment, but could also be shorter segments). Since the solutions for each height are computed independently, a matrix like the one just described is needed to solve for the vector velocities at each altitude sampled by the F region velocity experiment. [36] Acknowledgments. The Arecibo Observatory is operated by Cornell University through a cooperative agreement with the National Science Foundation. [37] Arthur Richmond thanks Richard Behnke and John M. Holt for their assistance in evaluating this paper.
References Aster, R. C., B. Borchers, and C. H. Thurber (2005), Parameter Estimation and Inverse Problems, Elsevier, New York. Burnside, R. G., J. C. G. Walker, and M. P. Sulzer (1987), Kinematic properties of the F region ion velocity field from incoherent scatter radar measurements at Arecibo, J. Geophys. Res., 92(A4), 3345 – 3355. Hagfors, T., and R. A. Behnke (1974), Measurements of three-dimensional plasma velocities at the Arecibo Observatory, Radio Sci., 9, 89 – 93. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1992), Numerical Recipes, Cambridge Univ. Press, New York. Sulzer, M. P. (1986), A phase modulation technique for a sevenfold statistical improvement in incoherent scatter data-taking, Radio Sci., 21, 737 – 744.
Regularization Matrix
[35] The H matrix in equation (18.5.12) of Press et al. [1992] is actually quite sparse and basically has two different vectors at the edges and a row vector at the middle
N. Aponte, S. A. Gonza´lez, and M. P. Sulzer, Arecibo Observatory, HC3 Box 53995, Arecibo, PR 00612. (
[email protected];
[email protected];
[email protected])
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